## Homework 5

1. At , a 1D harmonic oscillator is in a linear combination of the energy eigenstates

Find the expected value of as a function of time using operator methods.

2. Evaluate the uncertainty'' in for the 1D HO ground state where . Similarly, evaluate the uncertainty in for the ground state. What is the product ? Now do the same for the first excited state. What is the product for this state?

3. An operator is Unitary if . Prove that a unitary operator preserves inner products, that is . Show that if the states are orthonormal, that the states are also orthonormal. Show that if the states form a complete set, then the states also form a complete set.

4. Show at if an operator is hermitian, then the operator is unitary.

5. Calculate and .

6. Calculate by direct calculation. Now calculate the same thing using .

7. If is a polynomial in the operator , show that . As a result of this, note that since any energy eigenstate can be written as a series of raising operators times the ground state, we can represent by .

8. At a particle is in the one dimensional Harmonic Oscillator state .
• Compute the expected value of as a function of time using and in the Schrodinger picture.
• Now do the same in the Heisenberg picture.

Jim Branson 2013-04-22